The Liouville function in number-potential fields, derived following
standard methods
(see [1]}, [2]}, [3]}),
is
\(\hat{\mathcal {L}}& =\left( 1 - e^{\eta } \right)\left(\varepsilon + {\kappa }_1 x^2\right) +\left( 1 - e^{- \eta } \right)\left({\kappa }_2 + {\bar{\kappa }}_1 x^2\right) x .\) 
